Question: The quantity
\[\frac{\tan \frac{\pi}{5} + i}{\tan \frac{\pi}{5} - i}\]is a tenth root of unity.  In other words, it is equal to $\cos \frac{2n \pi}{10} + i \sin \frac{2n \pi}{10}$ for some integer $n$ between 0 and 9 inclusive.  Which value of $n$?
Solution: We have that
\begin{align*}
\frac{\tan \frac{\pi}{5} + i}{\tan \frac{\pi}{5} - i} &= \frac{\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}} + i}{\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}} - i} \\
&= \frac{\sin \frac{\pi}{5} + i \cos \frac{\pi}{5}}{\sin \frac{\pi}{5} - i \cos \frac{\pi}{5}} \\
&= \frac{i \sin \frac{\pi}{5} - \cos \frac{\pi}{5}}{i \sin \frac{\pi}{5} + \cos \frac{\pi}{5}} \\
&= \frac{\cos \frac{4 \pi}{5} + i \sin \frac{4 \pi}{5}}{\cos \frac{\pi}{5} + i \sin \frac{\pi}{5}} \\
&= \cos \frac{3 \pi}{5} + i \sin \frac{3 \pi}{5} \\
&= \cos \frac{6 \pi}{10} + i \sin \frac{6 \pi}{10}.
\end{align*}Thus, $n = \boxed{3}.$